Remarks on the Relation between Quantum and Classical Entropy as Proposed by A. Wehrl

According to Wehrl one associates to every state of a quantum system characterized by a density matrix o a “classical entropy” build up with help of the probability measure (zlwz)dz. Here (zl denotes the coherent state labelled by the point z of the phase space, and dz is the Liouville measure. In [l] and [2] Wehrl stated a number of interesting properties of this classical entropy and its relation to quantum entropy. In [3] Lieb proved SC1 > 1 in appropriate units (Boltzmann’s constant equal one), an inequality conjectured by Wehrl. In this note we begin with the technical remark that the mapo + (zloJz)dz is dual to a “quantization map” and can be easily extended to singular states. Then we comment on the fact that the mentioned map is mixing enhancing. From this, using Lieb’s inequality [3], we arrive at an a priori lower bound for F”’ where F is any concave function. We conclude with a further remark on singular states.